Cluster-Robust Variance Estimators: CRV 1-3

Introduction

summclust handles cluster robust variance estimation of linear regression models of the form

\[\begin{align} y &= X \beta + u \\ &= \begin{bmatrix} y_{1} \\ y_{2} \\ ...\\ y_{G} \end{bmatrix} = \begin{bmatrix} X_{1} \\ X_{2} \\ ...\\ X_{G} \end{bmatrix} \beta + \begin{bmatrix} u_{1} \\ u_{2} \\ ...\\ u_{G} \end{bmatrix}, \end{align}\]

with \(E(u|X) = 0\), where group \(g\) contains \(N_{g}\) observations so that \(N = \sum_{g = 1}^{G} N_{g}\). The regression residuals \(u\) are allowed to be correlated within clusters, but are assumed to be uncorrelated across clusters. In consequence, the models’ covariance matrix is block diagonal. For each cluster, we denote \(E(u_{g} u_{g}') =\Sigma_{g}\). We denote the predicted residual as \(\hat{u} = y - X \beta\).

Three Cluster-Robust Variance Estimators

The literature on cluster robust inference has proposed three different estimators, which all follow the same ‘sandwich’ structure

\[\begin{align} \hat{V}(\hat{\beta}) &= m (X'X)^{-1} (\sum_{g=1}^{G} \hat{\Sigma}_{g} ) (X'X)^{-1} \\ &= m (X'X)^{-1} (\sum_{g=1}^{G} \hat{s}_{g} \hat{s}_{g}') (X'X)^{-1} \end{align}\]

The three proposed types of CRV estimators vary in how \(\hat{s}_{g}\) is estimated and in how small sample corrections \(m\) are applied.

The most common cluster robust estimator, the CRV1 estimator, is defined as

\[\begin{align} \hat{s}_{g} = X_{g}'\hat{u}_{g} && \& && m = \frac{N-1}{N-k} / \frac{G}{G-1} \end{align}\]

When each cluster contains a unique observation, i.e. when \(G=N\), the CRV1 estimator is equivalent to the HC1 heteroskedasticity robust estimator.

The CRV2 estimator is computed as

\[\begin{align} \hat{s}_{g} = X_{g}' M_{gg}^{-1/2} \hat{u}_{g} && \& && m = 1. \end{align}\]

To define \(M_{gg}\), we first define the “hat” matrix

\[\begin{equation} H_g = X_g (X'X)^{-1} X_g' \end{equation}\]

and \(I_{N_g}\) as the diagonal matrix for all observations in cluster \(g\).

\(M_{gg}\) is then defined as

\[\begin{equation} M_{gg} = I_{N_g} - H_g. \end{equation}\]

Last, the CRV3 estimator is defined as

\[\begin{align} \hat{s}_{g} = X_{g}' M_{gg}^{-1} \hat{u}_{g} && \& && m = G/(G-1). \end{align}\]

Jackknife formulation of the CRV3 estimator (CRV3J)

Building on work by Niccodemi and Wansbeek, MacKinnon, Nielsen and Webb show that the CRV3 estimator can be computed as a Jackknife estimator.

First, let’s define \(\hat{\beta}^{(g)}\), the OLS estimate of (1) when cluster g is omitted:

\[\begin{equation} \hat{\beta}^{(g)} = ((X'X)^{-1} - (X_{g}'X_{g})^{-1})(X'y - X_{g}'y_{g}), g = 1, ... , G. \end{equation}\]

MNW show the the CRV3 estimator is equivalent to computing

\[\begin{equation} \hat{V}_{3}(\hat{\beta}) = \frac{G}{G-1} \sum{g = 1}^{G} (\hat{\beta}^{(g)} - \hat{\beta}) (\hat{\beta}^{(g)} - \hat{\beta})' \end{equation}\]

They further propose the following Jackknive estimator, CRVJ:

\[\begin{equation} \hat{V}_{3J}(\hat{\beta}) = \frac{G}{G-1} \sum{g = 1}^{G} (\hat{\beta}^{(g)} - \bar{\beta}) (\hat{\beta}^{(g)} - \bar{\beta})' \end{equation}\]

with \(\bar{\beta} = G^{-1} \sum_{g=1}^{G} \hat{\beta}^{(g)}\).

When \(G=N\), this Jackknife estimator is equivalent to the HC3 heteroskedasticity-robust estimator as proposed in MacKinnon and White.